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The Annals of Applied Statistics

2013, Vol. 7, No. 4, 2138–2156DOI: 10.1214/13-AOAS661c© Institute of Mathematical Statistics, 2013

BAYESIAN DATA AUGMENTATION DOSE FINDINGWITH CONTINUAL REASSESSMENT METHOD AND

DELAYED TOXICITY

By Suyu Liu∗, Guosheng Yin†,1 and Ying Yuan∗,2

MD Anderson Cancer Center∗ and University of Hong Kong†

A major practical impediment when implementing adaptive dose-finding designs is that the toxicity outcome used by the decision rulesmay not be observed shortly after the initiation of the treatment. Toaddress this issue, we propose the data augmentation continual re-assessment method (DA-CRM) for dose finding. By naturally treatingthe unobserved toxicities as missing data, we show that such missingdata are nonignorable in the sense that the missingness depends onthe unobserved outcomes. The Bayesian data augmentation approachis used to sample both the missing data and model parameters fromtheir posterior full conditional distributions. We evaluate the perfor-mance of the DA-CRM through extensive simulation studies and alsocompare it with other existing methods. The results show that theproposed design satisfactorily resolves the issues related to late-onsettoxicities and possesses desirable operating characteristics: treatingpatients more safely and also selecting the maximum tolerated dosewith a higher probability. The new DA-CRM is illustrated with twophase I cancer clinical trials.

1. Introduction. The continual reassessment method (CRM) proposedby O’Quigley, Pepe and Fisher (1990) is an influential phase I clinical trialdesign for finding the maximum tolerated dose (MTD) of a new drug. TheCRM assumes a single-parameter working dose–toxicity model and contin-uously updates the estimates of the toxicity probabilities of the considereddoses to guide dose escalation. Under some regularity conditions, the MTDidentified by the CRM generally converges to the true MTD, even when

Received February 2013; revised May 2013.1Supported in part by a Grant (784010) from the Research Grants Council of Hong

Kong.2Supported in part by the National Cancer Institute Grant R01CA154591-01A1.Key words and phrases. Bayesian adaptive design, late-onset toxicity, nonignorable

missing data, phase I clinical trial.

This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in The Annals of Applied Statistics,2013, Vol. 7, No. 4, 2138–2156. This reprint differs from the original in paginationand typographic detail.

1

2 S. LIU, G. YIN AND Y. YUAN

the working model is misspecified [Shen and O’Quigley (1996)]. A vari-ety of extensions of the CRM have been proposed to improve its practicalimplementation and operating characteristics [Goodman, Zahurak and Pi-antadosi (1995); Moller (1995); Heyd and Carlin (1999); Leung and Wang(2002); O’Quigley and Paoletti (2003); Garrett-Mayer (2006); Iasonos andO’Quigley (2011); among others]. Recently, several robust versions of theCRM have been proposed by using the Bayesian model averaging and poste-rior maximization [Yin and Yuan (2009) and Daimon, Zohar and O’Quigley(2011)], so that the method is insensitive to the prior specification of thedose–toxicity model.

In real applications, to achieve its best performance, the CRM requiresthat the toxicity outcome be observed quickly such that, by the time ofthe next dose assignment, the toxicity outcomes of the currently treatedpatients have been completely observed. However, late-onset toxicities arecommon in phase I clinical trials, especially in oncology areas. For example,in radiotherapy trials, dose-limiting toxicities (DLTs) often occur long afterthe treatment is finished [Coia, Myerson and Tepper (1995)]. Desai et al.(2007) reported a phase I study to determine the MTD of oxaliplatin forcombination with gemcitabine and the concurrent radiation therapy in pan-creatic cancer. In that trial, on average, a new patient arrived every twoweeks, whereas it took nine weeks to assess the toxicity outcomes of the pa-tients after the treatment is initiated. Consequently, at the moment of doseassignment for a newly arrived patient, the patients under treatment mightnot have yet completed the full assessment period and, thus, their toxicityoutcomes might not be available for making the decision of dose assignment.Late-onset toxicity has been becoming a more critical issue in the emergingera of the development of novel molecularly targeted agents, because manyof these agents tend to induce late-onset toxicities. A recent review paper inthe Journal of Clinical Oncology found that among a total of 445 patientsinvolved in 36 trials, 57% of the grade 3 and 4 toxicities were late-onset and,as a result, particular attention has been called upon the issue of late-onsettoxicity [Postel-Vinay et al. (2011)].

Our research is motivated by one of the collaborative projects, which in-volves the combination of chemo- and radiation therapy. The trial aims todetermine the MTD of a chemo-treatment, while the radiation therapy is de-livered as a simultaneous integrated boost in patients with locally advancedesophageal cancer. The DLT is defined as CTCAE 3.0 (Common Terminol-ogy Criteria for Adverse Events version 3.0) grade 3 or 4 esophagitis, andthe target toxicity rate is 30%. In this trial, six dose levels are investigatedand toxicity is expected to be late-onset. The accrual rate is approximately3 patients per month, but it generally takes 3 months to fully assess toxi-city for each patient. By the time of dose assignment for a newly enrolled

BAYESIAN DATA AUGMENTATION DOSE FINDING 3

patient, some patients who have not experienced toxicity thus far may ex-perience toxicity later during the remaining follow-up. It is worth notingthat whether we view toxicity as late-onset or not is relative to the patientaccrual rate. If patients enter the trial at a fast rate and toxicity evaluationcannot keep up with the speed of enrollment, this situation is considered aslate-onset toxicity. On the other hand, if the patient accrual is very slow,for example, one patient every three months, and toxicity evaluation also re-quires a follow-up of three months, then the trial conduct may not cause anymissing data problem. For broader applications besides this chemo-radiationtrial and to gain more insight into the missing data issue, we explore sev-eral options to design such late-onset toxicity trials, including the CRM andsome other possibilities discussed below.

Operatively, the CRM does not require that toxicity must be immediatelyobservable, and the update of posterior estimates and dose assignment canbe based on the currently observed toxicity data while ignoring the missingdata. However, such observed data represent a biased sample of the pop-ulation because patients who would experience toxicity are more likely tobe included in the sample than those who do not experience toxicity. Inother words, the observed data contain an excessively higher percentage oftoxicity than the complete data. Consequently, the estimates based on onlythe observed data tend to overestimate the toxicity probabilities and leadto overly conservative dose escalation. Alternatively, Cheung and Chappell(2000) proposed the time-to-event CRM (TITE-CRM), in which subjectswho have not experienced toxicity thus far are weighted by their follow-up times. Based on similar weighting methods, Braun (2006) studied bothearly- and late-onset toxicities in phase I trials; Mauguen, Le Deley andZohar (2011) investigated the EWOC design with incomplete toxicity data;and Wages, Conaway and O’Quigley (2013) proposed a dose-finding methodfor drug-combination trials. Yuan and Yin (2011) proposed an expectation–maximization (EM) CRM approach to handling late-onset toxicity.

In the Bayesian paradigm, we propose a data augmentation approachto resolving the late-onset toxicity problem based upon the missing datamethodology [Little and Rubin (2002); and Daniels and Hogan (2008)]. Bytreating the unobserved toxicity outcomes as missing data, we naturallyintegrate the missing data technique and theory into the CRM framework.In particular, we establish that the missing data due to late-onset toxicitiesare nonignorable. We propose the Bayesian data augmentation CRM (DA-CRM) to iteratively impute the missing data and sample from the posteriordistribution of the model parameters based on the imputed likelihood.

The remainder of the article is organized as follows. In Section 2 we brieflyreview the original CRM methodology and propose the DA-CRM based onBayesian data augmentation to address the missing data issue caused bylate-onset toxicity. In Section 3.1 we present simulation studies to compare

4 S. LIU, G. YIN AND Y. YUAN

the operating characteristics of the new design with other available methods,and in Section 3.2 we conduct a sensitivity analysis to further investigate theproperties of the DA-CRM. We illustrate the proposed DA-CRM design withtwo cancer clinical trials in Section 4, and conclude with a brief discussionin Section 5.

2. Dose-finding methods.

2.1. Continual reassessment method. In a phase I dose-finding trial, pa-tients enter the study sequentially and are followed for a fixed period oftime (0, T ) to assess the toxicity of the drug. During this evaluation window(0, T ), we measure a binary toxicity outcome for each subject i,

Yi =

{

1, if a drug-related toxicity is observed in (0, T ),

0, if no drug-related toxicity is observed in (0, T ).

Typically, the length of the assessment period T is chosen so that if a drug-related toxicity occurs, it would occur within (0, T ). Depending on the natureof the disease and the treatment agent, the assessment period T may varyfrom days to months.

Suppose that a set of J doses of a new drug are under investigation, theCRM assumes a working dose–toxicity curve, such as

πd = αexp(a)d , d= 1, . . . , J,

where πd is the true toxicity probability at dose level d, αd is the prespecifiedprobability constant, satisfying a monotonic dose–toxicity order α1 < · · ·<αJ , and a is an unknown parameter. We continuously update this dose–toxicity curve by re-estimating a based on the observed toxicity outcomesin the trial.

Suppose that n patients have entered the trial, and let yi and di denotethe binary toxicity outcome and the received dose level for the ith subject,respectively. The likelihood function based on the toxicity outcomes y ={yi, i= 1, . . . , n} is given by

L(y|a) =

n∏

i=1

{αexp(a)di

}yi{1− αexp(a)di

}1−yi .

Assuming a prior distribution f(a) for a, for example, f(a) is a normaldistribution with mean 0 and variance σ2, a∼N(0, σ2), then the posteriordistribution of a is given by

f(a|y) =L(y|a)f(a)

∫

L(y|a)f(a)da,(2.1)

BAYESIAN DATA AUGMENTATION DOSE FINDING 5

and the posterior means of the dose–toxicity probabilities are given by

πd =

∫

αexp(a)d f(a|y)da, d= 1, . . . , J.

Based on the updated estimates of the toxicity probabilities, the CRM as-signs a new cohort of patients to dose level d∗ which has an estimated toxicityprobability closest to the prespecified target φ, that is,

d∗ = argmind∈(1,...,J)

|πd − φ|.

The trial continues until the exhaustion of the total sample size, and thenthe dose with an estimated toxicity probability closest to φ is selected as theMTD.

2.2. Nonignorable missing data. One of the practical limitations of theCRM is that the DLT needs to be ascertainable quickly after the initiationof the treatment. Figure 1 illustrates the situation where the patient inter-arrival time τ is shorter than the assessment period T . By the time a dose isto be assigned to a newly accrued patient (say, patient 4 at time 3τ ), someof the patients who have entered the trial (i.e., patients 2 and 3) may have

Fig. 1. Illustration of missing toxicity outcomes under fast accrual. For each patient, thehorizontal line segment represents the follow-up, on which toxicity is indicated by a cross.At time τ , the toxicity outcome of patient 1 is missing (i.e., Y1 is missing); at time 2τ ,the toxicity outcome of patient 2 is missing (i.e., Y1 = 1, but Y2 is missing); and at time3τ , the toxicity outcomes of both patients 2 and 3 are missing (i.e., Y1 = 1, but Y2 and Y3

are missing).

6 S. LIU, G. YIN AND Y. YUAN

been partially followed and their toxicity outcomes are still not available.More precisely, for the ith subject, let ti denote the time to toxicity. Forsubjects who do not experience toxicity during the trial, we set ti =∞. Atthe moment of decision making for dose assignment, let ui (0≤ ui ≤ T ) de-note the actual follow-up time for subject i, and let Mi(ui) be the missingdata indicator for Yi. Then it follows that

Mi(ui) =

{

1, if ti > ui and ui < T ,

0, if ti ≤ ui or ui = T.(2.2)

That is, the toxicity outcome is missing with Mi(ui) = 1 for patients whohave not yet experienced toxicity (ti >ui) and have not been fully followedup to T (ui < T ); and the toxicity outcome is observed with Mi(ui) = 0when patients either have experienced toxicity (ti ≤ ui) or have completedthe entire follow-up (ui = T ) without experiencing toxicity. For notationalsimplicity, we suppress ui and take Mi ≡Mi(ui). Due to patients’ staggeredentry, it is reasonable to assume that ui is independent of ti, that is, thetime of dose assignment (or the arrival of a new patient) is independent ofthe time to toxicity.

Under the missing data mechanism (2.2), the induced missing data arenonignorable or informative because the probability of missingness of Yi

depends on the underlying time to toxicity, and thus implicitly depends onthe value of Yi itself. More specifically, the data from patients who wouldnot experience toxicity (Yi = 0) in the assessment period are more likely tobe missing than data from patients who would experience toxicity (Yi = 1).The next theorem provides a new insight to the issue of late-onset toxicity.

Theorem 1. Under the missing data mechanism (2.2), the missing datainduced by late-onset toxicity are nonignorable with Pr(Mi = 1|Yi = 0) >Pr(Mi = 1|Yi = 1).

The proof of the theorem is briefly sketched in the Appendix. In general,the missing data are more likely to occur for those patients who wouldnot experience toxicity in (0, T ). This phenomenon is also illustrated inFigure 1. Patient 2 who will not experience toxicity during the assessmentperiod is more likely to have a missing toxicity outcome at the decision-making times 2τ and 3τ than patient 1 who has experienced toxicity betweentimes τ and 2τ . Compared with other missing data mechanisms, such asmissing completely at random or missing at random, nonignorable missingdata are the most difficult to deal with [Little and Rubin (2002)], whichbrings a new challenge to clinical trial designs. Because the missing data arenonignorable, the naive approach by simply discarding the missing data andmaking dose-escalation decisions solely based on the observed toxicity datais problematic. The observed data represent a biased sample of the complete

BAYESIAN DATA AUGMENTATION DOSE FINDING 7

data and contain more toxicity observations than they should be becausethe responses for patients who would experience toxicity are more likely tobe observed. As a result, approaches based only on the observed toxicitydata typically overestimate the toxicity probabilities and thus lead to overlyconservative dose escalation.

During the trial conduct, the amount of missing data depends on the ratioof the assessment period T and the interarrival time τ , denoted as the A/Iratio = T/τ . The larger the value of the A/I ratio, the greater the amount ofmissing data that would be produced, because there would be more patientswho may not have completed the toxicity assessment when a new cohortarrives.

2.3. DA-CRM using Bayesian data augmentation. An intuitive approachto dealing with the unobserved toxicity outcomes is to impute the missingdata so that the standard complete-data method can be applied. One wayto achieve this goal is to use data augmentation (DA) proposed by Tannerand Wong (1987). The DA iterates between two steps: the imputation (I)step, in which the missing data are imputed, and the posterior (P) step,in which the posterior samples of unknown parameters are simulated basedon the imputed data. As the CRM is originally formulated in the Bayesianframework [O’Quigley, Pepe and Fisher (1990)], the DA provides a natu-ral and coherent way to address the missing data issue due to late-onsettoxicity. Note that the missing data we consider here is a special case ofnonignorable missing data with a known missing data mechanism as definedby (2.2). Therefore, the nonidentification problem that often plagues thenonignorable missing data can be circumvented as follows.

In order to obtain consistent estimates, we need to model the nonignorablemissing data mechanism in (2.2). Toward this goal, we specify a flexiblepiecewise exponential model for the time to toxicity for patients who wouldexperience DLTs, which concerns the conditional distribution of ti given Yi =1. Specifically, we consider a partition of the follow-up period [0, T ] into Kdisjoint intervals [0, h1), [h1, h2), . . . , [hK−1, hK ≡ T ], and assume a constanthazard λk in the kth interval. Define the observed time xi =min(ui, ti) andδik = 1 if the ith subject experiences toxicity in the kth interval; and δik = 0otherwise. Let λ = {λ1, . . . , λK}; when the toxicity data y = {y1, . . . , yn}are completely observed, the likelihood function of λ based on n enrolledsubjects is given by

L(y|λ) =n∏

i=1

K∏

k=1

λδikk exp{−yiλksik},

where sik = hk − hk−1 if xi > hk; sik = xi − hk−1 if xi ∈ [hk−1, hk); and oth-erwise sik = 0. Similar to the TITE-CRM, we assume that the time-to-DLT

8 S. LIU, G. YIN AND Y. YUAN

distribution is invariant to the dose level, conditioning on that the patientwill experience toxicity (Yi = 1). This assumption is helpful to pool informa-tion across different doses and obtain more reliable estimates. The sensitivityanalysis in Section 3.2 shows that our method is not sensitive to the violationof this assumption.

In the Bayesian paradigm, we assign each component of λ an independentgamma prior distribution with the shape parameter ζk and the rate param-eter ξk, denoted as Ga(ζk, ξk). When there is some prior knowledge availableregarding the shape of the hazard for the time to toxicity, the hyperparam-eters ζk and ξk can be calibrated to match the prior information. Here wefocus on the common case in which the prior information is vague and weaim to develop a default and automatic prior distribution for general use.Specifically, we assume that a priori toxicity occurs uniformly throughoutthe assessment period (0, T ), which represents a neutral prior opinion be-tween early-onset and late-onset toxicity. Under this assumption, the hazardat the middle of the kth partition is λk =K/{T (K − k + 0.5)}. Thus, weassign λk a gamma prior distribution,

λk =Ga(λk/C,1/C),

where C is a constant determining the size of the variance with respect tothe mean, as the mean for this prior distribution is λk and the variance isCλk. Based on our simulations, we found that C = 2 yields a reasonablyvague prior and equips the design with good operating characteristics.

Based on the time-to-toxicity model as above, the DA algorithm can beimplemented as follows. At the I step of the DA, we “impute” the missingdata by drawing posterior samples from their full conditional distributions.Let y = (yobs,ymis), where yobs and ymis denote the observed and missingtoxicity data, respectively; and let Dobs = (yobs,M) denote the observeddata with missing indicators M = {Mi, i = 1, . . . , n}. As the missing dataare informative, the observed data used for inference include not only theobserved toxicity outcomes yobs, but also the missing data indicators M.Inference that ignores M (such as the CRM) would lead to biased estimates.It can be shown that, conditional on the observed data Dobs and modelparameters (a,λ), the full conditional distribution of yi ∈ ymis is given by

yi|(Dobs, a,λ)∼ Bernoulli

(

αexp(a)di

exp(−∑K

k=1 λksik)

1− αexp(a)di

+ αexp(a)di

exp(−∑K

k=1λksik)

)

.

At the P step of the DA, given the imputed data y, we sequentially samplethe unknown model parameters from their full conditional distributions asfollows:

(i) Sample a from f(a|y) given by (2.1), where y is the “complete” dataafter filling in the missing outcomes.

BAYESIAN DATA AUGMENTATION DOSE FINDING 9

(ii) Sample λk, k = 1, . . . ,K, from

λk|y∼Ga

(

λk/C +

n∑

i=1

δik,1/C +

n∑

i=1

yisik

)

.

The DA procedure iteratively draws a sequence of samples of the missingdata and model parameters through the imputation (I) step and posterior(P) step until the Markov chain converges. The posterior samples of a canthen be used to make inference on πd to direct dose finding.

2.4. Dose-finding algorithm. Let φ denote the physician-specified toxic-ity target, and assume that patients are treated in cohorts, for example, witha cohort size of three. For safety, we restrict dose escalation or de-escalationby one dose level of change at a time. The dose-finding algorithm for theDA-CRM is described as follows:

(1) Patients in the first cohort are treated at the lowest dose level.(2) At the current dose level dcurr, based on the cumulated data, we obtain

the posterior means for the toxicity probabilities, πd (d= 1, . . . , J). We thenfind dose level d∗ that has a toxicity probability closest to φ, that is,

d∗ = argmind∈(1,...,J)

|πd − φ|.

• If dcurr > d∗, we de-escalate the dose level to dcurr − 1;• if dcurr < d∗, we escalate the dose level to dcurr +1;• otherwise, the dose stays at the same level as dcurr for the next cohort of

patients.

(3) Once the maximum sample size is reached, the dose that has thetoxicity probability closest to φ is selected as the MTD.

In addition, we also impose an early stopping rule for safety: if Pr(π1 >φ|Dobs) > 0.96, the trial will be terminated. That is, if the lowest dose isstill overly toxic, the trial should be stopped early.

3. Numerical studies.

3.1. Simulations. To examine the operating characteristics of the DA-CRM design, we conducted extensive simulation studies. We considered sixdose levels and assumed that toxicity monotonically increased with respectto the dose. The target toxicity probability was 30% and a maximum num-ber of 12 cohorts were treated sequentially in a cohort size of three. Thesample size was chosen to match the maximum sample size required bythe conventional “3 + 3” design. The toxicity assessment period was T = 3

10 S. LIU, G. YIN AND Y. YUAN

months and the accrual rate was 6 patients per month. That is, the inter-arrival time between every two consecutive cohorts was τ = 0.5 month withthe A/I ratio = 6.

We considered four toxicity scenarios in which the MTD was located atdifferent dose levels. Due to the limitation of space, we show only scenarios1 and 2 in Table 1, and the other scenarios are provided in Table S1 ofthe supplementary materials [Liu, Yin and Yuan (2013)]. Under each sce-nario, we simulated times to toxicity based on Weibull, log-logistic and uni-form distributions, respectively. For Weibull and log-logistic distributions,we controlled that 70% toxicity events would occur in the latter half of theassessment period (T/2, T ). Specifically, at each dose level, the scale andshape parameters of the Weibull distribution were chosen such that

(1) the cumulative distribution function at the end of the follow-up timeT would be the toxicity probability of that dose; and

(2) among all the toxicities that occurred in (0, T ), 70% of them wouldoccur in (T/2, T ), the latter half of the assessment period.

Because the toxicity probability varies across different dose levels, the scaleand shape parameters of the Weibull distribution need to be carefully chosenfor different dose levels, and similarly for the scale and location parametersof the log-logistic distribution. For the uniform distribution, we simulatedthe time to toxicity independently for each dose level and controlled thecumulative distribution function at the end of the follow-up time T match-ing the toxicity probability of each dose. In the proposed DA-CRM, weusedK = 9 partitions to construct the piecewise exponential time-to-toxicitymodel. We compared the DA-CRM with the CRMobs, which determined thedose assignment based on only the observed toxicity data as suggested byO’Quigley, Pepe and Fisher (1990), and the TITE-CRM with the adaptiveweighting scheme proposed by Cheung and Chappell (2000). As a bench-mark for comparison, we also implemented the complete-data version of theCRM (denoted by CRMcomp), assuming that all of the toxicity outcomesin the trial were completely observed prior to each dose assignment. TheCRMcomp required repeatedly suspending the accrual prior to each doseassignment to wait that all of the toxicity outcomes in the trial were com-pletely observed. Although the CRMcomp is not feasible in practice whentoxicities are late-onset, it provides an optimal upper bound to evaluate theperformances of other designs. Actually, when all toxicity outcomes are ob-servable (i.e., no missing data), the DA-CRM and TITE-CRM are equivalentto the complete-data CRMcomp. For all methods, we set the probability con-stants in the CRM (α1, . . . , α6) = (0.08,0.12,0.20,0.30,0.40, 0.50) and useda normal prior distribution N(0,2) for parameter a. Under each scenario,we simulated 5000 trials.

BAYESIAN DATA AUGMENTATION DOSE FINDING 11

Table 1

Simulation study comparing the complete-data CRM (CRMcomp), the CRM based on theobserved toxicity data only (CRMobs), time-to-event CRM (TITE-CRM) and the

proposed data augmentation CRM (DA-CRM) with the sample size 36, the cohort size 3and the A/I ratio = 6

Recommendation percentage at dose levelTime totoxicity Design 1 2 3 4 5 6 None NMTD+

Duration(months)

Scenario 1 Pr(toxicity) 0.1 0.15 0.3 0.45 0.6 0.7CRMcomp 0.6 13.8 61.9 22.9 0.6 0.0 0.2 9.0 36.4# patients 4.8 7.2 14.9 7.6 1.3 0.1

Weibull CRMobs 0.4 7.5 48.4 27.4 2.4 0.1 13.7 4.0 8.2# patients 7.5 8.3 13.2 3.0 0.9 0.1TITE-CRM 3.4 23.1 55.9 16.5 0.6 0.0 0.5 15.5 9.0# patients 5.1 6.3 9.0 8.1 4.9 2.5DA-CRM 0.9 14.7 56.4 25.1 1.5 0.0 1.2 10.4 8.9# patients 9.4 7.6 8.3 6.0 3.0 1.3

Log-logistic CRMobs 0.3 7.8 48.2 27.4 2.7 0.1 13.6 4.0 8.2# patients 7.4 8.4 13.2 3.0 0.9 0.1TITE-CRM 3.6 22.6 56.3 16.5 0.6 0.0 0.4 15.4 9.0# patients 5.1 6.4 9.1 8.2 4.8 2.4DA-CRM 0.9 13.9 58.1 23.8 1.9 0.0 1.3 10.3 8.9# patients 9.5 7.6 8.2 6.0 3.0 1.3

Uniform CRMobs 0.1 4.7 38.0 30.5 2.9 0.1 23.6 2.8 7.5# patients 8.6 8.2 10.9 2.3 0.5 0.0TITE-CRM 2.1 20.4 56.6 19.1 1.4 0.0 0.4 13.0 9.0# patients 6.3 7.0 9.7 8.0 3.7 1.3DA-CRM 0.6 11.2 56.9 27.6 1.7 0.0 1.9 8.7 8.9# patients 10.8 7.6 8.5 5.7 2.2 0.7

Scenario 2 Pr(toxicity) 0.08 0.1 0.2 0.3 0.45 0.6CRMcomp 0.0 1.4 23 55.9 18.8 0.8 0.1 6.6 36.4# patients 4.1 4.1 9.0 12.2 5.5 1.0

Weibull CRMobs 0.0 1.0 18.9 48.5 22.2 1.8 7.6 2.9 8.5# patients 5.7 6.7 13.8 5.2 2.3 0.7TITE-CRM 0.1 2.6 29.2 52.4 14.9 0.8 0.1 11.2 9.0# patients 4.2 4.7 7.1 8.8 6.8 4.4DA-CRM 0.0 1.5 24.4 54.0 17.7 1.4 1.1 7.3 8.9# patients 8.4 6.3 7.0 6.7 4.5 2.8

Log-logistic CRMobs 0.0 0.9 19.0 48.5 22.0 1.9 7.7 2.9 8.5# patients 5.7 6.7 13.8 5.1 2.3 0.7TITE-CRM 0.1 2.6 29.0 52.3 15.1 0.8 0.1 11.1 9.0# patients 4.1 4.7 7.2 8.8 6.8 4.3DA-CRM 0.0 1.4 23.3 54.0 18.7 1.6 1.0 7.5 8.9# patients 8.3 6.2 6.9 6.8 4.6 2.9

Uniform CRMobs 0.0 0.7 14.9 45.3 23.3 2.3 13.5 2.0 8.1# patients 6.9 7.1 12.3 4.5 1.7 0.4TITE-CRM 0.1 2.2 26.9 54.4 15.4 0.9 0.1 9.6 9.0# patients 4.9 5.0 7.6 8.8 6.2 3.3DA-CRM 0.0 1.3 23.7 54.0 18.8 1.1 1.2 6.2 8.9# patients 9.3 6.3 7.1 6.8 4.2 2.0

12 S. LIU, G. YIN AND Y. YUAN

Following each scenario in Tables 1 and S1, the first row is the true tox-icity probabilities; rows 2 and 3 show the dose selection probability (withthe percentage of inconclusive trials denoted by “None”) and the averagenumber of patients treated at each dose based on the complete-data de-sign CRMcomp, respectively; the remaining rows provide the correspondingsummary statistics for the CRMobs, TITE-CRM and DA-CRM under var-ious settings of late-onset toxicity and time-to-toxicity distributions. TheCRMcomp does not depend on the distributions of the times to toxicity be-cause the design assumes that all toxicity outcomes are completely observedbefore each dose assignment.

When evaluating the trial designs with late-onset toxicities, one of themost important measures of the design performance is patient safety becausethe main issue of the late-onset toxicities is that ignoring them will leadto overly aggressive dose escalation and thus treating too many patientsat excessively toxic doses, that is, the doses higher than the MTD. As ameasure of safety, in Tables 1 and S1, we also report the number of patientstreated at doses above the MTD (denoted as NMTD+) averaged across 5000simulated trials.

In scenario 1, the MTD (shown in boldface) is at dose level 3, and thecomplete-data design CRMcomp yielded an optimal selection probability of61.9%. The selection probability of the MTD using the DA-CRM was slightlylower than this optimal value, but higher than that of using the CRMobs.For instance, when the time to toxicity followed the log-logistic distribution,the selection probability using the DA-CRM was 58.1%, whereas that of theCRMobs was 48.2%. The CRMobs appeared to be overly conservative and ledto a high percentage (about 13.6%) of inconclusive trials. This was becausethe CRMobs estimated the toxicity probabilities based solely on the observedtoxicity data, which is a biased sample of the complete data with an excessivenumber of toxicities. Therefore, the CRMobs tended to overestimate thetoxicity probabilities, resulting in conservative dose escalations and highpercentages of early termination of the trial. The TITE-CRM yielded similarselection percentages as the DA-CRM, but the DA-CRM was much safer:the number of patients treated above the MTD (i.e., NMTD+) using the DA-CRM was notably smaller than that of the TITE-CRM and close to that ofthe complete-data design. For example, when the time to toxicity followedthe Weibull distribution, NMTD+ was 9.0 and 10.4 using the complete-datadesign and the DA-CRM, respectively, while that based on the TITE-CRMwas 15.5. As the CRMobs is overly conservative, NMTD+ = 4.0 is the smallestunder the CRMobs.

In scenario 2, the MTD is the fourth dose, and in scenario 3 (see TableS1), the MTD is the second. Compared to the TITE-CRM, the DA-CRMyielded comparable MTD-selection probabilities but appeared to be safer,which reduced NMTD+ by more than 30% in both scenarios. For example,

BAYESIAN DATA AUGMENTATION DOSE FINDING 13

in scenario 2, when the time to toxicity followed the Weibull distribution,NMTD+ using the DA-CRM was 7.3, approximately 35% less than that usingthe TITE-CRM (NMTD+ = 11.2). A similar extent of decreasing in NMTD+

was observed in scenario 3 when using the DA-CRM. The CRMobs again ledto a high percentage of inconclusive trials (particularly under the uniformdistribution) and a relatively low selection percentage of the MTD due tothe overestimation of the toxicity probabilities. For scenario 4 in Table S1,in which the fifth dose is the MTD, the CRMobs yielded a similar selectionpercentage as the TITE-CRM and DA-CRM.

We further investigated the performance of the designs under a smallersample size of 27 patients treated in a cohort size of 3, and 21 patientstreated in a cohort size of 1. The pattern of the results is generally similar tothose described above (see Tables S2 and S3 in the supplementary materials[Liu, Yin and Yuan (2013)]). We also examined the operating characteristicsof the DA-CRM under a lower A/I ratio of 3 with the cohort interarrivaltime τ = 1 month (see Table S4 in the supplementary materials [Liu, Yinand Yuan (2013)]). In this case, the accrual rate was relatively slower and,thus, late-onset toxicities became of less concern since the majority of tox-icity outcomes would be observed at the moment of dose assignment. Asexpected, the performances of the CRMobs, TITE-CRM and DA-CRM wererather comparable across different scenarios and time-toxicity distributions.Actually, when the A/I ratio is less than or equal to 1 (i.e., no late-onsettoxicities and no missing data), the CRMobs, TITE-CRM and DA-CRM areexactly the same.

These results suggest that, when the A/I ratio is low (e.g., when thedisease under study is rare and thus the accrual rate is slow), the CRMobs haslittle bias and is still a good design option for phase I clinical trials. However,when the accrual is fast, for example, in multi-center clinical trials for somecommon type of cancer (e.g., breast or lung cancer), the A/I ratio can be high(particularly when radiotherapies or some targeted agents are used), andusing the proposed DA-CRM can lead to better operating characteristics.

3.2. Sensitivity analysis. We investigated the robustness of the proposedDA-CRM design when (1) the underlying times to toxicity were heteroge-neous across the doses, by simulating the times to toxicity from a Weibulldistribution at dose levels of 1, 3 and 5, and from a log-logistic distribu-tion at dose levels of 2, 4 and 6; (2) the number of partitions used in thepiecewise exponential model for the times to toxicity was K = 5 and 12; and(3) the prior distribution for a was N(0,0.57), the “least-informative” priorproposed by Lee and Cheung (2011). The results show that the performanceof the DA-CRM (e.g., the selection percentages and NMTD+) was very sim-ilar across different conditions (see Table S5 in the supplementary materials[Liu, Yin and Yuan (2013)]), which suggests the robustness of the proposeddesign.

14 S. LIU, G. YIN AND Y. YUAN

Table 2

Days, doses and DLTs for eighteen evaluable patients enrolled in the pancreatic cancertrial

Patient Day on Day off Dose Patient Day on Day off DoseNo. study∗ study∗ (mg/m2) DLT No. study∗ study∗ (mg/m2) DLT

1 0 67 30 No 10 224 291 50 No2 43 98 30 No 11 280 303 50 Yes3 50 116 30 No 12 301 347 50 Yes4 56 108 30 No 13 322 382 50 No5 70 133 40 No 14 329 389 50 No6 147 217 40 No 15 343 372 50 Yes7 161 224 40 No 16 364 423 40 No8 182 238 40 No 17 371 408 50 Yes9 224 284 50 No 18 455 528 30 No

* days since the initiation of the study.

4. Applications.

4.1. Pancreatic cancer trial. Muler et al. (2004) described a phase I trialto determine the MTD of cisplatin that could be added to the full-dosegemcitabine and radiation therapy in patients with pancreatic cancer. Theprotocol treatment consisted of two 28-day cycles of chemotherapy, withradiation given during the first cycle of chemotherapy. Radiation and gem-citabine doses were held constant, while four dose levels of cisplatin (20, 30,40 and 50 mg/m2) were investigated in the trial. The DLTs were defined asCTCAE 2.0 grade 4 thrombocytopenia, grade 4 neutropenia lasting morethan 7 days or grade 3 toxicity in other organ systems. Patients were requiredto be followed for nine weeks in order to fully assess their toxicity outcomes.The goal of the trial was to determine the dose of cisplatin associated witha target DLT rate of 20%.

As shown in Table 2, one challenge of designing this trial is that theaccrual was fast, compared to the 9-week assessment period for DLTs (i.e.,the toxicity was late-onset). In the DA-CRM design, we took α= (0.1,0.15,0.2,0.25) as the prior estimates of the toxicity probabilities for the four doselevels of cisplatin, and used 30 mg/m2 as the starting dose of the trial.For patient safety, we required that at least two patients must have fullycompleted their toxicity assessment at the lower dose before the dose can beescalated to the next higher level.

Figure 2 summarizes how the posterior estimates of the toxicity proba-bilities of four doses were updated as more patients were enrolled duringthe trial conduct. The first four patients were assigned to the dose of 30mg/m2. Based on the days from the initiation of the trial, when patient 5arrived on day 70, patient 1 had completed the follow-up, while patients 2,

BAYESIAN DATA AUGMENTATION DOSE FINDING 15

Fig. 2. Estimates of the toxicity probabilities of four doses with the cumulative numberof patients in the pancreatic cancer trial. Numbers 1–4 in the figure indicate the four doselevels.

3 and 4 had finished only 43%, 32% and 22% of their follow-ups, withoutexperiencing any DLTs. The estimates of toxicity probabilities of four doseswere π = (0.113,0.131,0.148,0.165). We escalated the dose and subsequentlytreated patients 5 to 8 at the dose of 40 mg/m2. Patient 2 died after 63 dayson therapy, but was judged to be secondary to the hypercoaguable stateassociated with pancreatic cancer. Therefore, that death was classified asunrelated to therapy (i.e., not a DLT).

Upon the arrival of patient 9 on day 224, patients 1 to 7 had completedtheir toxicity assessment and none of them had experienced DLT. Thesedata yielded the updated toxicity estimates π = (0.005,0.008,0.013,0.019),suggesting that the doses of 30 mg/m2 and 40 mg/m2 were safe. As a result,we further escalated the dose and assigned patients 9 to 11 to 50 mg/m2.On day 301 when patient 12 was accrued, patients 1 to 10 had completedtheir toxicity assessment without experiencing DLT, yielding the updatedestimates of the toxicity probabilities π = (0.007,0.012,0.019,0.027). Conse-quently, patients 12 to 15 were also treated at 50 mg/m2.

After patient 12 experienced a DLT (i.e., duodenal ulcer) on day 347,the estimates of the toxicity probabilities began to increase, that is, π =(0.085,0.125,0.165,0.207), but not sufficiently to trigger dose de-escalation.According to the dose-finding algorithm, the incoming patients 16 and 17

16 S. LIU, G. YIN AND Y. YUAN

should be treated at 50 mg/m2. However, because the investigators wereconcerned about a potential DLT in patient 15, only patient 17 was treatedat 50 mg/m2, while patient 16 was treated at a lower dose 40 mg/m2.

By the time that patient 18, the last enrolled patient, arrived on day 455, 4out of 8 patients previously treated at 50 mg/m2 had experienced DLTs (i.e.,one duodenal ulcer, one diarrhea resulting in dehydration, and two grade 3anorexia and nausea leading to a two-level decline in performance status).This significantly increased π to (0.126, 0.177, 0.228, 0.275). Therefore, pa-tient 18 was assigned to a lower dose 30 mg/m2. At the end of the trial, theestimates of the toxicity probabilities were π = (0.118,0.167,0.215,0.264)and, thus, the dose 40 mg/m2 was selected as the MTD because its esti-mated toxicity probability was closest to the target of 0.2.

4.2. Esophageal cancer trial. In the esophageal cancer clinical trial de-scribed in Introduction, the target toxicity probability was 30% and a totalof 30 patients were treated sequentially in cohorts of size 3. Six doses wereinvestigated and the trial started by treating the first cohort at dose level1. Under the DA-CRM design, the posterior estimates of the dose–toxicityprobabilities were updated only when the first patient of each new cohort(i.e., patient 4, 7, 10, 13, 16, . . . ) was enrolled.

The three patients in cohort 1 were enrolled at days 3, 6 and 18, respec-tively (see Table S6 in the supplementary materials [Liu, Yin and Yuan(2013)]). On day 28 when patient 4 (i.e., the first patient of cohort 2)was enrolled, the three patients in cohort 1 had finished only 28%, 24%and 11% of their 3-month follow-ups without experiencing toxicity (i.e.,DLT). The estimates of the toxicity probabilities of six dose levels were π =(0.172,0.185,0.209,0.236,0.264,0.294). We escalated the dose and treatedpatient 4, and subsequently patients 5 and 6, at dose level 2.

When patient 7 (the first patient of cohort 3) arrived on day 57, weagain updated the estimates of the toxicity probabilities and obtained π =(0.315,0.336,0.369,0.407,0.445,0.486). Although at that moment we stillhad not observed any DLTs yet, the values of π increased compared withthe previous estimates of π. This is because on day 57, more patients (i.e.,patients 1 to 6) were under treatment and none of them had finished their3-month follow-ups yet. There was greater uncertainty regarding the toxicityprobabilities of the doses and it was preferable to be conservative. Our al-gorithm automatically took into account such uncertainty and de-escalatedthe dose back to the first level for treating cohort 3.

On day 91 when the first patient of cohort 4 (i.e., patient 10) was accrued,patients 1, 2 and 3 were very close to completing their 3-month follow-upswithout experiencing toxicity, indicating that the first dose level was safeand dose escalation was needed. The proposed algorithm timely reflectedthis data information and escalated the dose to level 2. The dose was further

BAYESIAN DATA AUGMENTATION DOSE FINDING 17

Fig. 3. Estimate of the unknown parameter a with cumulative cohorts in the esophagealcancer trial.

escalated to levels 3 and 4 for treating cohorts 5 and 6, respectively, as noDLT was observed. By the time patient 19 arrived, the toxicity outcomes ofall patients treated in the trial had been observed. In particular, all threepatients (i.e., patients 16–18) treated at dose level 4 had experienced DLTs.Our algorithm de-escalated the dose to level 3 to treat cohort 7. Thereafter,there were always at least 18 toxicity outcomes (from patients 1–18) fullyobserved, thus, π became rather stable and consistently indicated that dose3 was the MTD. The last 3 cohorts were all treated at dose level 3 and, atthe end of the trial, dose 3 was selected as the MTD with the estimatedtoxicity probability of 0.259.

Figure 3 displays the estimate of the unknown parameter a during thetrial conduct. At the beginning of the trial, there was much variability for theestimate of a due to sparse data, while the estimate became stabilized aftersix cohorts of patients were enrolled. Correspondingly, Table 3 summarizesthe estimates of the toxicity probabilities π for the six doses at each decision-making time.

5. Conclusions. We have proposed the DA-CRM design to address theissues associated with late-onset toxicities in phase I dose-finding trials. Inthe new design, unobserved toxicity outcomes are naturally treated as miss-ing data. We established that such missing data are nonignorable and linked

18 S. LIU, G. YIN AND Y. YUAN

Table 3

Estimates of the toxicity probabilities of six doses with the cumulative number of cohortsin the esophageal cancer trial

Cumulative number of cohorts

Dose level 1 2 3 4 5 6 7 8 9 10

1 0.172 0.315 0.028 0.021 0.080 0.175 0.186 0.171 0.189 0.1252 0.185 0.336 0.035 0.026 0.114 0.228 0.240 0.223 0.243 0.1723 0.209 0.369 0.050 0.037 0.181 0.320 0.333 0.314 0.337 0.2594 0.236 0.407 0.071 0.053 0.268 0.422 0.435 0.415 0.440 0.3615 0.264 0.445 0.096 0.071 0.359 0.515 0.528 0.509 0.532 0.4586 0.294 0.486 0.128 0.093 0.454 0.603 0.615 0.598 0.619 0.553

the missing data mechanism with the time to toxicity based on a flexiblepiecewise exponential model. Simulation studies showed that the DA-CRMoutperforms other available methods, particularly when toxicities need along follow-up time to be assessed. The selection percentage of the DA-CRM is often close to the optimal value, and many fewer patients would betreated at overly toxic doses.

This paper has focused on the single-agent dose finding using the CRM,but the proposed methodology provides a general and systematic approachto transforming the late-onset toxicity problem into a standard complete-data problem by imputing the missing toxicity outcomes. The proposedmethod can serve as a universal adaptor to extend existing trial designsto accommodate more complicated dose-finding problems with late-onsettoxicity. For example, by incorporating the data augmentation procedureinto the partial-order CRM [Wages, Conaway and O’Quigley (2011)], wecan address the late-onset toxicity for drug-combination trials or dose findingwith group heterogeneity. It is also worth emphasizing that although we havefocused on the late-onset toxicity, the proposed method can also be used tohandle other kinds of late-onset outcomes, such as delayed efficacy responsesin phase I/II or phase II trials, as well as response-adaptive randomizationdesigns.

APPENDIX: PROOF OF THEOREM 1

Considering that each subject is fully followed up to T , if ti > T , thenYi = 0; and if ti ≤ T , then Yi = 1. We demonstrate the nonignorable miss-ingness for the missing data caused by late-onset toxicity as follows. For asubject who will not experience toxicity, the probability that his/her toxicityoutcome will be missing is given by

Pr(Mi = 1|Yi = 0) = Pr(ti > ui, ui < T |Yi = 0)

BAYESIAN DATA AUGMENTATION DOSE FINDING 19

=Pr(ui < T |Yi = 0)Pr(ti >ui|ui < T,Yi = 0)

= Pr(ui < T |ti > T )Pr(ti > ui|ui < T, ti >T )

= Pr(ui < T ),

where the last equality follows because ti and ui are independent, and Pr(ti >ui|ui < T, ti > T ) = 1. Similarly, for a subject who will experience toxicity,the probability that his/her toxicity outcome will be missing is given by

Pr(Mi = 1|Yi = 1) = Pr(ti > ui, ui < T |Yi = 1)

= Pr(ui < T |Yi = 1)Pr(ti >ui|ui < T,Yi = 1)

= Pr(ui < T |ti ≤ T )Pr(ti > ui|ui < T, ti ≤ T )

= Pr(ui < T )Pr(ti >ui|ui < T, ti ≤ T ).

Because of Pr(ti > ui|ui < T, ti ≤ T )< 1, it follows that

Pr(Mi = 1|Yi = 0)>Pr(Mi = 1|Yi = 1).

Therefore, the missing data are more likely to occur for those patients whowill not experience toxicity in (0, T ).

Acknowledgments. The authors thank the Editor Professor Kafadar, As-sociate Editor and three anonymous referees for their many constructive andinsightful comments that have resulted in significant improvements in thearticle.

SUPPLEMENTARY MATERIAL

Additional simulation results (DOI: 10.1214/13-AOAS661SUPP; .pdf).Additional simulation results.

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S. Liu

Y. Yuan

Department of Biostatistics

University of Texas

MD Anderson Cancer Center

Houston, Texas 77030

USA

E-mail: syliu@mdanderson.orgyyuan@mdanderson.org

G. Yin

Department of Statistics

and Actuarial Science

University of Hong Kong

Hong Kong

China

E-mail: gyin@hku.hk